Comedies of manners are plays which are "set in the world of the upper class, and ridicule the pretensions of those who consider themselves socially superior, deflating them with satire. With witty dialogue and cleverly constructed scenarios, comedies of manners comment on the standards and mores of society and explore the relationships of the sexes. " (enotes Literary Criticism). The play Pygmalion is many things, and one of those is certainly a comedy of manners.
Pygmalion is less of a farce than the earlier comedies of manners. While there are certainly miscommunications (especially at the beginning between Eliza and Higgins) they generally don't create changes to the plot. But in several other respects Pygmalion is a comedy of manners; most especially in its satire of both the upper and lower classes.
Higgins, though a respected figure and "professor of phonetics" is an example of much of what is wrong, in Shaw's mind, with the upper class of Britain -- particularly the upper class intellectual male (a class to which Shaw, himself, belonged!) Higgins is a master of his subject, and, as he boasts, is not only able to place with minute accuracy London accents, but is able to train the most entrenched Cockney accents (Eliza's) into the tones of Mayfair. But with all his learning and accomplishments Higgins has left his humanity behind. He has to be forced (with only limited success, by Mrs. Pearce, Eliza, and Colonel Pickering) to understand that Eliza (as well as other people) is a human being like himself. Higgins has been duped by his own intellectual success into thinking less about human beings and becoming lost in his subject. Already of a peremptory and impatient character, this makes Higgins almost insufferable. The lack of malice in his actions does not entirely make up for his thoughtlessness and unintentional cruelty. Higgins, the man of the mind, has lost his soul. It could be construed that this is a commentary on British (male) erudition in general.
The silliness of Freddy Eynsford Hill, and, though he is poor, his unfitness for work is another commentary on the British upper class -- and it is not a favorable commentary. The impractical education and refinement of the poverty-stricken gentry (the Eynsford Hills' two-word last name is a marker of their illustrious, possibly noble descent) during Shaw's time, even when faced with a real decline into poverty, struck Shaw as ridiculous. That Freddy should have been trained to go out and make his living in some fashion is a definite subtext in the plot; as it is, Freddy has been trained all his life to be nothing more than a useless drone, with perfect manners and ease in drawing rooms which he will never be able to afford himself.
There is commentary, too, on the lower classes. The sponging, drunken father of Eliza, Alfred, is cheerful, bluff, and well-liked, but he essentially works the system. He even calls himself the "undeserving" poor (who nevertheless needs charity as much as the deserving poor -- a doubtlessly true statement) and is no virtuous paragon of the downtrodden underclass. The ugliness of poverty is not dressed up in Pygmalion; it manifests itself in crime, addiction, despair, and abuse (as it always has). This is not the usual province of a comedy of manners, but since the intersection of the two classes forms the plot of the play, Shaw could hardly neglect it.